Rotation Driven by Position - Driver Issue When Rendering
This is a bug. The Drivers - at least in this case and at least in 2.82 - update only if the current frame is changed manually by user. If it is changed by script (which Render Animation also is) Drivers ignore it. You can check it by typing in Blender Console C.scene.frame_current = 5 - or any other frame number - and you'll see, that your object rotation stays unchanged though your frame has changed. More to say, for example if you are at the frame 5, manually change frame to 4 and than back to 5 - the result on the frame 5 will be different than if you would change frame to 6 and than back to 5.

Blender Drivers are buggy. I'd suggest to use Animation Nodes addon instead, it works more stable in general.